Explicit functorial correspondences for level zero representations of p-adic linear groups

Let F be a non-Archimedean local field and D a central F-division algebra of dimension n², n?1. We consider first the irreducible smooth representations of D^× trivial on 1-units, and second the indecomposable, n-dimensional, semisimple, Weil-Deligne representations of F which are trivial on wild inertia. The sets of equivalence classes of these two sorts of representations are in canonical (functorial) bijection via the composition of the Jacquet-Langlands correspondence and the Langlands correspondence. They are also in canonical bijection via explicit parametrizations in terms of tame admissible pairs. This paper gives the relation between these two bijections. It is based on analysis of the discrete series of the general linear group GLn(F) in terms of a classification by extended simple types.     

Diabatic limit, eta invariants and Cauchy-Riemann manifolds of dimension 3

We relate a recently introduced non-local invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various η-invariants: on the one hand a renormalized η-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the η-invariant of the middle degree operator of the contact complex. We then provide explicit computations for transverse circle invariant CR structures on Seifert manifolds. This yields obstructions to filling a CR manifold by complex hyperbolic, Kähler-Einstein, or Einstein manifolds.     

On the existence of local smooth repulsive stabilizing feedbacks in dimension three

Given an affine control system in R³ subject to the Hörmander's condition at the origin, we prove the existence of a local smooth repulsive stabilizing feedback at the origin. Our construction is based on the classical homogenization procedure, on the existence of a semiconcave control-Lyapunov function, and on the classification of singularities of semiconcave functions in dimension two.     

Central limit theorem for the heat kernel measure on the unitary group

We prove that for a finite collection of real-valued functions f₁,…,fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of under the properly scaled heat kernel measure at a given time on the unitary group U(N) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N−1. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.     

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion

In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a $Q$-Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.     

Hecke group algebras as quotients of affine Hecke algebras at level 0

The Hecke group algebra of a finite Coxeter group , as introduced by the first and last authors, is obtained from by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, is the natural quotient of the affine Hecke algebra H(W)(q) through its level 0 representation.The proof relies on the following core combinatorial result: at level 0 the 0-Hecke algebra H(W)(0) acts transitively on . Equivalently, in type A, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level 0 representation is a calibrated principal series representation M(t) for a suitable choice of character t, so that the quotient factors (non-trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the 0-Hecke algebra and that of the affine Hecke algebra H(W)(q) at this specialization.     

Carre symetrique d'une forme primitive

A direct proof is given for the functionnal equation satisfied by the primitive symmetric square attached to a newform of any level. The method uses technics of modular forms theory and is based on a carefully study of minimal forms and Atkin-Lehner operators on twisted forms. The root number is explicit thanks to some properties of Gauss sums averaes analogous to Barnes Lemma.     

On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions

In this paper, we deal with a two-dimensional Navier-Stokes system in a rectangle with Navier slip boundary conditions on the horizontal sides. We establish the global null controllability of the system by controlling the normal component and the vorticity of the velocity on the vertical sides. The linearized control system around zero is controllable but one does not know how to deduce global controllability results for the nonlinear system. Our proof uses the return method together with a local exact controllability result by Fursikov and Imanuvilov.